The reductive-groups tag has no wiki summary.

**11**

votes

**2**answers

645 views

### Is there a scheme parametrizing the closed subgroups of an algebraic group?

In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...

**5**

votes

**1**answer

319 views

### Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...

**0**

votes

**0**answers

51 views

### Convergence of Selberg type Zeta function

Let $X=G/K$ be a Riemannian symmetric space without compact factor. We can assume $G$ is connected and real reductive. $K$ is a maximal subgroup of $G$.
Let $\Gamma$ be a discrete torsion free ...

**1**

vote

**0**answers

77 views

### Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers.
Can one classify all reductive group schemes over $C$?
Certainly, you have the trivial ones (coming from pulling-back ...

**2**

votes

**2**answers

196 views

### Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$

Let $K$ be a global field and set $O := \prod_{v\nmid \infty} O_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}_n(O) = \prod_v \mathrm{GL}_n(O_v)$ with the product of the $v$-adic ...

**2**

votes

**0**answers

96 views

### $X$-points of reductive group schemes, if $X$ is a proper smooth curve over a finite field

Let $X$ be a connected proper smooth curve over a finite field (so the generic point of $X$ is the spectrum of a global field $K$), and let $G \rightarrow X$ be an affine $X$-group scheme of finite ...

**2**

votes

**1**answer

149 views

### Unipotent conjugacy classes

Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?

**0**

votes

**1**answer

61 views

### Decomposition of Lie subspaces

If $M=G/H$ is a reductive homogeneous space then we can write $\frak{g}=\frak{m}+\frak{h}$
where $[\frak{h}, \frak{m}]\subset \frak{m}$. Here $\frak{g}$ and $\frak{h}$ are the Lie algebras of $G$ and ...

**-1**

votes

**1**answer

213 views

### Decomposition of $S^7=Spin(7)/G_2$

The seven sphere can be written as the reductive space $S^7=Spin(7)/G_2$. Has the decomposition $Spin(7)=G_2\times S^7$ been calculated somewhere; maybe in terms of Cayley numbers?

**4**

votes

**0**answers

147 views

### Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...

**1**

vote

**1**answer

167 views

### $\Gamma$-action on maximal tori in Borel-Tits

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not ...

**3**

votes

**1**answer

169 views

### Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus

Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center.
Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$.
Let $\chi : T(\mathbf{Q}_p) \to ...

**1**

vote

**0**answers

77 views

### Splitting for Subsequence of Automorphism Sequence for Algebraic Groups

Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups
$$
1\to \mathrm{Inn}(G)\to ...

**6**

votes

**0**answers

126 views

### Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...

**1**

vote

**1**answer

67 views

### Strictly contracting elements in the center of a Levi subgroup

Let $G$ be a connected reductive group over a non archimedean local field $k$.
Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M ...

**1**

vote

**0**answers

122 views

### Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...

**3**

votes

**1**answer

129 views

### Supercuspidal with Iwahori fixed vector

Let $F$ be a local field. Is there a reference for the following fact:
No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector?
I have a proof, by I'd prefer a reference, ...

**0**

votes

**1**answer

72 views

### Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...

**3**

votes

**1**answer

215 views

### What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...

**1**

vote

**1**answer

153 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

**0**

votes

**1**answer

165 views

### number of simple representations

For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? ...

**1**

vote

**0**answers

147 views

### Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...

**8**

votes

**1**answer

407 views

### Why people usually consider reductive groups in GIT?

Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT?
I found that the property of completely reducible of ...

**6**

votes

**2**answers

265 views

### Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...

**7**

votes

**1**answer

220 views

### Open cell decomposition after applying a Weyl group element

Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For ...

**2**

votes

**0**answers

84 views

### Invariant vectors in supercuspidal representations of GL_2(Zp)

Let $o$ be the ring of integers in a local field $F$ with prime-ideal $p$. Let $K$ be either $GL_2(o)$ or the normalizer of the Iwahori subgroup. Let $\sigma$ be a representation of $K$ times the ...

**3**

votes

**1**answer

137 views

### Semisimple group not split by an unramified extension?

Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?

**6**

votes

**2**answers

445 views

### Reference request: expository text on the structure of reductive groups over non-archimedean local fields

I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner ...

**3**

votes

**0**answers

145 views

### How to think about non-connected reductive groups

Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits.
Is ...

**4**

votes

**1**answer

278 views

### How to translate the representation theory of semisimple to reductive groups?

I am aware of the following question: Definitions of Reductive and Semisimple Groups
So let me phrase a precise question:
Is there a standard technique by which one can translate the ...

**5**

votes

**2**answers

204 views

### Principal series of finite group of Lie type

I have a naive question on complex representations of finite groups of Lie type.
Let $\bf G$ be a reductive group (say connected, with connected center, for safety)
defined over a finite field ...

**3**

votes

**0**answers

132 views

### Decomposition of k-split tori of p-adic reductive groups

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$.
There is a group homomorphism :
...

**2**

votes

**1**answer

250 views

### Weyl group of the restriction of scalars of split reductive group

Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$.
Set $G' = ...

**4**

votes

**1**answer

180 views

### Steinberg reps of reductive groups over local fields vs finite fields

Let $G$ be a reductive group over a non-archimedean field $F$ with reisdue field $f$.
Edit: The statements only make sense modulo tensoring by one-dimensional representations.
Are the unitary, ...

**2**

votes

**2**answers

204 views

### a conjugacy question in quasi-split reductive groups

I have a somewhat technical question about conjugacy in
quasi-reductive groups.
Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected
quasi-split reductive group over ...

**21**

votes

**1**answer

1k views

### Reconciling Lusztig's results with the Langlands philosophy

Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = ...

**21**

votes

**1**answer

734 views

### What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...

**3**

votes

**1**answer

229 views

### Volume of PGL(2,F) \ PGL(2, A)

Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...

**2**

votes

**0**answers

149 views

### Do there exist pseudo-reductive (but not reductive) groups of small dimension?

I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...

**10**

votes

**4**answers

908 views

### Is the normalizer of a reductive subgroup reductive?

Let $G$ be a reductive algebraic group over an algebraically closed field (of characteristic zero if it matters) and $H \subset G$ a subgroup, also reductive. Is the identity component of the ...

**4**

votes

**1**answer

263 views

### Does there exist a categorical treatment of root data(systems)?

What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, ...

**4**

votes

**3**answers

318 views

### Are all irreducible supercuspidal representation induced from compact-mod-center subgroups?

Let $G$ be a reductive group over a local non-archimedean field $F$.
Can every irreducible supercuspidal representation of $G(F)$ be realized as the induction from an open subgroup, which is compact ...

**2**

votes

**2**answers

355 views

### description of an endomorphism algebra

Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).
I ...

**7**

votes

**2**answers

524 views

### commuting elements in a reductive group

Does anyone know if the following holds?
Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.
To make things easier, you ...

**2**

votes

**1**answer

252 views

### Pseudo coefficients and orbital integrals

I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained:
"We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...

**1**

vote

**1**answer

205 views

### Heights in reductive groups

Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...

**3**

votes

**1**answer

161 views

### homogenous bundles

Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I ...

**1**

vote

**1**answer

395 views

### moduli problem for flag varieties?

Hi,
Suppose $G$ is a reductive group over an algebraiclly closed field $k$
(suppose $k$ of char zero if you want at first). Let $X$ be its flag variety.
Question: What is the moduli problem that $X$ ...

**1**

vote

**1**answer

230 views

### Intertwining Integral defined on a Weyl group?

Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group?
...

**2**

votes

**1**answer

442 views

### When is compact induction in GL(2) from an open compact group admissible?

Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional ...